A bungee jumper of mass of 60.0 kg falls off a bridge with zero initial velocity. The bridge is 50.0 m above the river. When the cord is stretched at its maximum (position B), the cord stores 10.0 kJ of elastic potential energy. Neglect air friction. Calculate the final height of the jumper relative to the river.
PE = mgh = 10kJ
you want 50-h
A bungee jumper of mass of 60.0 kg falls off a bridge with zero initial velocity. The bridge is 50.0 m above the river. When the cord is stretched at its maximum (position B), the cord stores 10.0 kJ of elastic potential energy. Neglect air friction.
(a)Express conservation of energy between A and B using
symbols
To calculate the final height of the jumper relative to the river, we need to consider the conservation of mechanical energy.
The total mechanical energy of the system (jumper + Earth) is conserved, so the initial mechanical energy at position A (the bridge) is equal to the final mechanical energy at position B (the maximum stretch of the cord).
The initial mechanical energy at position A consists of the gravitational potential energy of the jumper:
Initial Mechanical Energy (A) = mgh
Where:
m = mass of the jumper = 60.0 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height of the bridge above the river = 50.0 m
Initial Mechanical Energy (A) = (60.0 kg) * (9.8 m/s^2) * (50.0 m) = 29400 J
The final mechanical energy at position B consists of the gravitational potential energy and the elastic potential energy stored in the cord:
Final Mechanical Energy (B) = mgh + elastic potential energy
Where:
elastic potential energy = 10.0 kJ = 10.0 * 1000 J = 10000 J
Final Mechanical Energy (B) = (60.0 kg) * (9.8 m/s^2) * (h) + 10000 J
Since the jumper reaches the maximum stretch of the cord, the maximum kinetic energy is zero. Therefore, the final mechanical energy at position B consists only of the gravitational potential energy and the elastic potential energy.
Setting the initial mechanical energy equal to the final mechanical energy:
Initial Mechanical Energy (A) = Final Mechanical Energy (B)
29400 J = (60.0 kg) * (9.8 m/s^2) * (h) + 10000 J
Rearranging the equation to solve for h:
(60.0 kg) * (9.8 m/s^2) * (h) = 29400 J - 10000 J
(60.0 kg) * (9.8 m/s^2) * (h) = 19400 J
h = 19400 J / [(60.0 kg) * (9.8 m/s^2)]
h ≈ 32.4 m
Therefore, the final height of the jumper relative to the river is approximately 32.4 meters.
To find the final height of the jumper relative to the river, we need to consider the conservation of energy. The initial energy of the system will be solely potential energy, and the final energy of the system will be a combination of potential and kinetic energy.
1. Calculate the initial potential energy of the jumper:
Given that the jumper has a mass of 60.0 kg and the height of the bridge is 50.0 m, we can use the equation for potential energy:
Potential Energy (PE) = mass (m) * acceleration due to gravity (g) * height (h)
PE_initial = 60.0 kg * 9.8 m/s^2 * 50.0 m
2. Calculate the final energy of the system:
At the maximum stretch (position B) of the bungee cord, a portion of the initial potential energy is stored as elastic potential energy. The remaining energy is converted into kinetic energy.
Elastic Potential Energy (EPE) at position B = 10.0 kJ = 10,000 J (since 1 kJ = 1,000 J)
Kinetic Energy (KE) at position B = PE_initial - EPE
3. Calculate the final height:
At the final height, the kinetic energy will be zero, and all the initial potential energy will be converted back into gravitational potential energy.
Final Potential Energy (PE_final) = PE_initial - KE
Therefore, the final height (h_final) can be calculated using the equation for potential energy:
h_final = PE_final / (m * g)
Substituting the values we have calculated, we can find the final height of the jumper relative to the river.